Simply stated:
A phoneme is the smallest piece of sound that can be separately spoken in a word.
A phoneme is similar to "a letter", but many letters have more than one phoneme. The letter "A" for example has four phonemes -- the 'A' like 'cake', the 'A' like 'cat', the 'A' like 'father', and the 'A' like 'ago'. For someone who has never heard a sound and isn't oral, this will probably seem strange, but hearing people say those four 'A' words differently.
English has approximately 42 - 45 phonemes. When we're learning to read, we are sometimes taught that there are 5 vowels, but there are actually almost 20 'vowel sounds' or phonemes. For some reason, the consonant letters don't have an many variations in their phonemes in English. They usually have only one sound.
Other languages have more or fewer phonemes. The Paupa New Guinea language of 'Rotokas' only has 11 phonemes. The Botswanan language of "Xóõ" has 112 phonemes!
Hope that helps!